Results for 'lattice-ordered abelian groups'

995 found
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  1.  22
    Lattice-ordered Abelian groups and perfect mv-algebras: A topos-theoretic perspective.Olivia Caramello & Anna Carla Russo - 2016 - Bulletin of Symbolic Logic 22 (2):170-214.
    We establish, generalizing Di Nola and Lettieri’s categorical equivalence, a Morita-equivalence between the theory of lattice-ordered abelian groups and that of perfect MV-algebras. Further, after observing that the two theories are not bi-interpretable in the classical sense, we identify, by considering appropriate topos-theoretic invariants on their common classifying topos, three levels of bi-interpretability holding for particular classes of formulas: irreducible formulas, geometric sentences, and imaginaries. Lastly, by investigating the classifying topos of the theory of perfect MV-algebras, (...)
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  2.  19
    Free abelian lattice-ordered groups.A. M. W. Glass, Angus Macintyre & Françoise Point - 2005 - Annals of Pure and Applied Logic 134 (2-3):265-283.
    Let n be a positive integer and FAℓ be the free abelian lattice-ordered group on n generators. We prove that FAℓ and FAℓ do not satisfy the same first-order sentences in the language if m≠n. We also show that is decidable iff n{1,2}. Finally, we apply a similar analysis and get analogous results for the free finitely generated vector lattices.
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  3.  18
    Model-completions for Abelian lattice-ordered groups with finitely many disjoint elements.Philip Scowcroft - 2019 - Annals of Pure and Applied Logic 170 (6):673-698.
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  4.  28
    Erratum to “Free abelian lattice-ordered groups” [Ann. Pure Appl. Logic 134 (2–3) (2005) 265–283].A. M. W. Glass, Angus Macintyre & Françoise Point - 2016 - Annals of Pure and Applied Logic 167 (4):431-433.
  5.  24
    The logic of equilibrium and abelian lattice ordered groups.Adriana Galli, Renato A. Lewin & Marta Sagastume - 2004 - Archive for Mathematical Logic 43 (2):141-158.
    We introduce a deductive system Bal which models the logic of balance of opposing forces or of balance between conflicting evidence or influences. ‘‘Truth values’’ are interpreted as deviations from a state of equilibrium, so in this sense, the theorems of Bal are to be interpreted as balanced statements, for which reason there is only one distinguished truth value, namely the one that represents equilibrium. The main results are that the system Bal is algebraizable in the sense of [5] and (...)
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  6.  96
    Abelian Logic and the Logics of Pointed Lattice-Ordered Varieties.Francesco Paoli, Matthew Spinks & Robert Veroff - 2008 - Logica Universalis 2 (2):209-233.
    We consider the class of pointed varieties of algebras having a lattice term reduct and we show that each such variety gives rise in a natural way, and according to a regular pattern, to at least three interesting logics. Although the mentioned class includes several logically and algebraically significant examples (e.g. Boolean algebras, MV algebras, Boolean algebras with operators, residuated lattices and their subvarieties, algebras from quantum logic or from depth relevant logic), we consider here in greater detail (...) ℓ-groups, where such logics respectively correspond to: i) Meyer and Slaney’s Abelian logic [31]; ii) Galli et al.’s logic of equilibrium [21]; iii) a new logic of “preservation of truth degrees”. (shrink)
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  7.  16
    Corrigendum to “Model-completions for Abelian lattice-ordered groups with finitely many disjoint elements” [Ann. Pure Appl. Logic 170 (2019) 673–698]. [REVIEW]Philip Scowcroft - 2019 - Annals of Pure and Applied Logic 170 (11):102720.
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  8.  41
    An elementary presentation of the equivalence between MV-algebras and l-groups with strong unit.Roberto Cignoli & Daniele Mundici - 1998 - Studia Logica 61 (1):49-64.
    Aim of this paper is to provide a self-contained presentation of the natural equivalence between MV-algebras and lattice-ordered abelian groups with strong unit.
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  9.  19
    Boolean Skeletons of MV-algebras and ℓ-groups.Roberto Cignoli - 2011 - Studia Logica 98 (1-2):141-147.
    Let Γ be Mundici’s functor from the category $${\mathcal{LG}}$$ whose objects are the lattice-ordered abelian groups ( ℓ -groups for short) with a distinguished strong order unit and the morphisms are the unital homomorphisms, onto the category $${\mathcal{MV}}$$ of MV-algebras and homomorphisms. It is shown that for each strong order unit u of an ℓ -group G , the Boolean skeleton of the MV-algebra Γ ( G , u ) is isomorphic to the Boolean algebra (...)
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  10. Ordered Abelian-group products and inclusions studied in the 1st-order.F. Delon & F. Lucas - 1989 - Journal of Symbolic Logic 54 (2):499-511.
  11.  12
    Linear Abelian Modal Logic.Hamzeh Mohammadi - 2024 - Bulletin of the Section of Logic 53 (1):1-28.
    A many-valued modal logic, called linear abelian modal logic \(\rm {\mathbf{LK(A)}}\) is introduced as an extension of the abelian modal logic \(\rm \mathbf{K(A)}\). Abelian modal logic \(\rm \mathbf{K(A)}\) is the minimal modal extension of the logic of lattice-ordered abelian groups. The logic \(\rm \mathbf{LK(A)}\) is axiomatized by extending \(\rm \mathbf{K(A)}\) with the modal axiom schemas \(\Box(\varphi\vee\psi)\rightarrow(\Box\varphi\vee\Box\psi)\) and \((\Box\varphi\wedge\Box\psi)\rightarrow\Box(\varphi\wedge\psi)\). Completeness theorem with respect to algebraic semantics and a hypersequent calculus admitting cut-elimination are established. Finally, (...)
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  12.  9
    Elimination of Imaginaries in Ordered Abelian Groups with Bounded Regular Rank.Mariana Vicaría - 2023 - Journal of Symbolic Logic 88 (4):1639-1654.
    In this paper we study elimination of imaginaries in some classes of pure ordered abelian groups. For the class of ordered abelian groups with bounded regular rank (equivalently with finite spines) we obtain weak elimination of imaginaries once we add sorts for the quotient groups $\Gamma /\Delta $ for each definable convex subgroup $\Delta $, and sorts for the quotient groups $\Gamma /(\Delta + \ell \Gamma )$ where $\Delta $ is a definable (...)
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  13.  15
    Existential equivalence of ordered abelian groups with parameters.V. Weispfenning - 1990 - Archive for Mathematical Logic 29 (4):237-248.
    In [GK], Gurevich and Kokorin proved that any two non-trivial ordered abelian groups (o-groups, for short) satisfy the same existential sentences. Let nowG, H be non-trivialo-groups with a commono-subgroupG 0. We determine whetherG andH are existentially equivalent overG 0. As a corollary, we obtain algebraic criteria for deciding, whether ano-subgroupG is existentially closed in ano-groupH. Corresponding results are proved foro-groups in which congruences are regarded as atomic relations.
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  14.  7
    Expanded theory of ordered Abelian groups.Yuri Gurevich - 1977 - Annals of Mathematical Logic 12 (2):193-228.
  15.  35
    Given a divisible ordered abelian group Λ, we call (X, d) a Λ-metric space if d: X× X−→ Λ satisfies the usual axioms of a metric, ie, for all x, y∈ X, d (x, y)− d (y, x)≥ 0 if and only if x= y, and the triangle inequality holds. We can now give the definition of asymptotic cone according to van den Dries and Wilkie.Linus Kramer & Katrin Tent - 2004 - Bulletin of Symbolic Logic 10 (2):175-185.
    §1. Introduction. Asymptotic cones of metric spaces were first invented by Gromov. They are metric spaces which capture the ‘large-scale structure’ of the underlying metric space. Later, van den Dries and Wilkie gave a more general construction of asymptotic cones using ultrapowers. Certain facts about asymptotic cones, like the completeness of the metric space, now follow rather easily from saturation properties of ultrapowers, and in this survey, we want to present two applications of the van den Dries-Wilkie approach. Using ultrapowers (...)
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  16.  14
    The Complexity of Bounded Quantifiers in Some Ordered Abelian Groups.Philip Scowcroft - 2007 - Notre Dame Journal of Formal Logic 48 (4):521-550.
    This paper obtains lower and upper bounds for the number of alternations of bounded quantifiers needed to express all formulas in certain ordered Abelian groups admitting elimination of unbounded quantifiers. The paper also establishes model-theoretic tests for equivalence to a formula with a given number of alternations of bounded quantifiers.
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  17.  18
    An undecidability theorem for lattices over group rings.Carlo Toffalori - 1997 - Annals of Pure and Applied Logic 88 (2-3):241-262.
    Let G be a finite group, T denote the theory of Z[G]-lattices . It is shown that T is undecidable when there are a prime p and a p-subgroup S of G such that S is cyclic of order p4, or p is odd and S is non-cyclic of order p2, or p = 2 and S is a non-cyclic abelian group of order 8 . More precisely, first we prove that T is undecidable because it interprets the word (...)
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  18.  7
    Model Completions for Universal Classes of Algebras: Necessary and Sufficient Conditions.George Metcalfe & Luca Reggio - 2023 - Journal of Symbolic Logic 88 (1):381-417.
    Necessary and sufficient conditions are presented for the (first-order) theory of a universal class of algebraic structures (algebras) to have a model completion, extending a characterization provided by Wheeler. For varieties of algebras that have equationally definable principal congruences and the compact intersection property, these conditions yield a more elegant characterization obtained (in a slightly more restricted setting) by Ghilardi and Zawadowski. Moreover, it is shown that under certain further assumptions on congruence lattices, the existence of a model completion implies (...)
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  19.  13
    Arithmetic of Dedekind cuts of ordered Abelian groups.Antongiulio Fornasiero & Marcello Mamino - 2008 - Annals of Pure and Applied Logic 156 (2):210-244.
    We study Dedekind cuts on ordered Abelian groups. We introduce a monoid structure on them, and we characterise, via a suitable representation theorem, the universal part of the theory of such structures.
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  20.  10
    Topological properties of definable sets in ordered Abelian groups of burden 2.Alfred Dolich & John Goodrick - 2023 - Mathematical Logic Quarterly 69 (2):147-164.
    We obtain some new results on the topology of unary definable sets in expansions of densely ordered Abelian groups of burden 2. In the special case in which the structure has dp‐rank 2, we show that the existence of an infinite definable discrete set precludes the definability of a set which is dense and codense in an interval, or of a set which is topologically like the Cantor middle‐third set (Theorem 2.9). If it has burden 2 and (...)
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  21.  13
    Lattice-ordered reduced special groups.M. Dickmann, M. Marshall & F. Miraglia - 2005 - Annals of Pure and Applied Logic 132 (1):27-49.
    Special groups [M. Dickmann, F. Miraglia, Special Groups : Boolean-Theoretic Methods in the Theory of Quadratic Forms, Memoirs Amer. Math. Soc., vol. 689, Amer. Math. Soc., Providence, RI, 2000] are a first-order axiomatization of the theory of quadratic forms. In Section 2 we investigate reduced special groups which are a lattice under their natural representation partial order ; we show that this lattice property is preserved under most of the standard constructions on RSGs; in particular (...)
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  22.  16
    The order indiscernibles of divisible ordered Abelian groups.David Rosenthal - 1984 - Journal of Symbolic Logic 49 (1):151-160.
  23.  40
    The elementary class of products of totally ordered Abelian groups.Daniel Gluschankof - 1991 - Journal of Symbolic Logic 56 (1):295-299.
  24.  16
    A characterization of MV-algebras free over finite distributive lattices.Vincenzo Marra - 2008 - Archive for Mathematical Logic 47 (3):263-276.
    Mundici has recently established a characterization of free finitely generated MV-algebras similar in spirit to the representation of the free Boolean algebra with a countably infinite set of free generators as any Boolean algebra that is countable and atomless. No reference to universal properties is made in either theorem. Our main result is an extension of Mundici’s theorem to the whole class of MV-algebras that are free over some finite distributive lattice.
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  25.  6
    Decidability Results for Classes of Ordered Abelian Groups in Logics with Ramsey-Quantifiers.Wolfgang Lenski - 1995 - In M. Krynicki, M. Mostowski & L. Szczerba (eds.), Quantifiers: Logics, Models and Computation. Kluwer Academic Publishers. pp. 139--168.
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  26.  12
    Conical logic and l-groups logic.Marta S. Sagastume - 2005 - Journal of Applied Non-Classical Logics 15 (3):265-283.
    It is well known that there is a categorical equivalence between lattice-ordered Abelian groups (or l-groups) and conical BCK-algebras (see [COR 80]). The aim of this paper is to study this equivalence from the perspective of logic, in particular, to study the relationship between two deductive systems: conical logic Co and a logic of l-groups, Balo. In [GAL 04] the authors introduce a system Bal which models the logic of balance of opposing forces with (...)
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  27.  7
    Abraham Robinson and Elias Zakon. Elementary properties of ordered abelian groups. Transactions of the American Mathematical Society, vol. 96 , pp. 222–236. [REVIEW]J. Mennicke - 1967 - Journal of Symbolic Logic 32 (3):414.
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  28. Review: Abraham Robinson, Elias Zakon, Elementary Properties of Ordered Abelian Groups[REVIEW]J. Mennicke - 1967 - Journal of Symbolic Logic 32 (3):414-414.
  29.  16
    The Boolean algebras of Abelian groups and well-orders.Dale Myers - 1974 - Journal of Symbolic Logic 39 (3):452-458.
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  30. Hyper-regular lattice-ordered groups.Daniel Gluschankof & François Lucas - 1993 - Journal of Symbolic Logic 58 (4):1342-1358.
  31.  35
    The Lattice of Subvarieties of the Variety Defined by Externally Compatible Identities of Abelian Groups of Exponent n.Katarzyna Gajewska-Kurdziel & Krystyna Mruczek-Nasieniewska - 2007 - Studia Logica 85 (3):361-379.
    The lattices of varieties were studied in many works (see [4], [5], [11], [24], [31]). In this paper we describe the lattice of all subvarieties of the variety $G_{Ex}^n$ defined by so called externally compatible identities of Abelian groups and the identity xⁿ ≈ yxⁿ. The notation in this paper is the same as in [2].
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  32.  19
    Proof theory for lattice-ordered groups.Nikolaos Galatos & George Metcalfe - 2016 - Annals of Pure and Applied Logic 167 (8):707-724.
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  33. Degrees of orders on torsion-free Abelian groups.Asher M. Kach, Karen Lange & Reed Solomon - 2013 - Annals of Pure and Applied Logic 164 (7-8):822-836.
    We show that if H is an effectively completely decomposable computable torsion-free abelian group, then there is a computable copy G of H such that G has computable orders but not orders of every degree.
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  34. P-compatible Abelian groups.Krystyna Mruczek-Nasieniewska - 2005 - Logic and Logical Philosophy 14 (2):253-263.
    Let τ : F → N be a type of a variety V . Every partition Pof the set F determines a so-called P-compatible variety. We consider thevarieties GnP defined by so-called P-compatible identities of Abelian groupswith exponent n. Besides, we study a connection between the lattice of allpartitions of the set F and the lattice of all subvarieties of the variety definedby some kind of P-compatible identities — externally compatible identitiessatisfied in the class of all (...) groups with exponent n. (shrink)
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  35.  32
    Externally compatible Abelian groups of the type (2,1,0).Krystyna Mruczek-Nasieniewska - 2006 - Logic and Logical Philosophy 15 (3):239-250.
    In [4] the lattice of all subvarieties of the variety G n Ex defined by so called externally compatible identities of Abelian groups together with the identity x n ≈ y n , for any n ∈ N and n ≥ 1 was described. In that paper classes of models of the type (2,1) where considered. It appears that diagrams of lattices of subvariaties defined by externally compatible identities satisfied in a given equational theory depend on the (...)
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  36.  43
    A classification of certain group-like FL $$_e$$ e -chains.Sándor Jenei & Franco Montagna - 2015 - Synthese 192 (7):2095-2121.
    Classification of certain group-like FL $_e$ -chains is given: We define absorbent-continuity of FL $_e$ -algebras, along with the notion of subreal chains, and classify absorbent-continuous, group-like FL $_e$ -algebras over subreal chains: The algebra is determined by its negative cone, and the negative cone can only be chosen from a certain subclass of BL-chains, namely, one with components which are either cancellative (that is, those components are negative cones of totally ordered Abelian groups) or two-element MV-algebras, (...)
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  37.  25
    Polynomial-time abelian groups.Douglas Cenzer & Jeffrey Remmel - 1992 - Annals of Pure and Applied Logic 56 (1-3):313-363.
    This paper is a continuation of the authors' work , where the main problem considered was whether a given recursive structure is recursively isomorphic to a polynomial-time structure. In that paper, a recursive Abelian group was constructed which is not recursively isomorphic to any polynomial-time Abelian group. We now show that if every element of a recursive Abelian group has finite order, then the group is recursively isomorphic to a polynomial-time group. Furthermore, if the orders are bounded, (...)
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  38.  22
    Uniqueness of the implication for totally ordered MV-algebras.Néstor G. Martı́nez & Alejandro Petrovich - 2001 - Annals of Pure and Applied Logic 108 (1-3):261-268.
    It is shown that in a linearly ordered MV-algebra A , the implication is unique if and only if the identity function is the unique De Morgan automorphism on A . Modulo categorical equivalence, our uniqueness criterion recalls Ohkuma's rigidness condition for totally ordered abelian groups. We also show that, if A is an Archimedean totally ordered MV-algebra, then each non-trivial De Morgan automorphism of the underlying involutive lattice of A yields a new implication (...)
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  39.  19
    Relative categoricity in abelian groups II.Wilfrid Hodges & Anatoly Yakovlev - 2009 - Annals of Pure and Applied Logic 158 (3):203-231.
    We consider structures A consisting of an abelian group with a subgroup AP distinguished by a 1-ary relation symbol P, and complete theories T of such structures. Such a theory T is -categorical if T has models A of cardinality λ with AP=κ, and given any two such models A,B with AP=BP, there is an isomorphism from A to B which is the identity on AP. We classify all complete theories of such structures A in terms of the cardinal (...)
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  40.  10
    The left adjoint of Spec from a category of lattice-ordered groups.José Luis Castiglioni & Hernán Javier San Martín - 2016 - Journal of Applied Logic 15:1-15.
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  41.  24
    The representation theorem for the algebras determined by the fragments of infinite-valued logic of Lukasiewicz.Barbara Wozniakowska - 1978 - Bulletin of the Section of Logic 7 (4):176-178.
    In this paper we shall give a characterization of D-algebras in terms of lattice ordered abelian groups. To make this paper self-contained we shall recall some notations from [4]. The symbols !; ^; _; serve as implication, conjunction, disjunction, and negation, respectively. By D we mean a set of connectives from the list above containing the implication connective !. By a D-formula we mean a formula built up in a usual way from an innite set of (...)
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  42.  20
    Logic and computation, Proceedings of a workshop held at Carnegie Mellon University, June 30–July 2, 1987, edited by Wilfried Sieg, Contemporary Mathematics, vol. 106, American Mathematical Society, Providence1990, xiv + 297 pp. - Douglas K. Brown. Notions of closed subsets of a complete separable metric space in weak subsystems of second order arithmetic. Pp. 39–50. - Kostas Hatzikiriakou and Stephen G. Simpson. WKL0 and orderings of countable abelian groups. Pp. 177–180. - Jeffry L. Hirst. Marriage theorems and reverse mathematics. Pp. 181–196. - Xiaokang Yu. Radon–Nikodym theorem is equivalent to arithmetical comprehension. Pp. 289–297. - Fernando Ferreira. Polynomial time computable arithmetic. Pp. 137–156. - Wilfried Buchholz and Wilfried Sieg. A note on polynomial time computable arithmetic. Pp. 51–55. - Samuel R. Buss. Axiomatizations and conservation results for fragments of bounded arithmetic. Pp. 57–84. - Gaisi Takeuti. Sharply bounded arithmetic and the function a – 1. Pp. 2. [REVIEW]Jörg Hudelmaier - 1996 - Journal of Symbolic Logic 61 (2):697-699.
  43.  40
    A simplified duality for implicative lattices and l-groups.Nestor G. Martinez - 1996 - Studia Logica 56 (1-2):185 - 204.
    A topological duality is presented for a wide class of lattice-ordered structures including lattice-ordered groups. In this new approach, which simplifies considerably previous results of the author, the dual space is obtained by endowing the Priestley space of the underlying lattice with two binary functions, linked by set-theoretical complement and acting as symmetrical partners. In the particular case of l-groups, one of these functions is the usual product of sets and the axiomatization of (...)
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  44.  50
    Type-Decomposition of an Effect Algebra.David J. Foulis & Sylvia Pulmannová - 2010 - Foundations of Physics 40 (9-10):1543-1565.
    Effect algebras (EAs), play a significant role in quantum logic, are featured in the theory of partially ordered Abelian groups, and generalize orthoalgebras, MV-algebras, orthomodular posets, orthomodular lattices, modular ortholattices, and boolean algebras.We study centrally orthocomplete effect algebras (COEAs), i.e., EAs satisfying the condition that every family of elements that is dominated by an orthogonal family of central elements has a supremum. For COEAs, we introduce a general notion of decomposition into types; prove that a COEA factors (...)
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  45.  11
    The Hahn Embedding Theorem for a Class of Residuated Semigroups.Sándor Jenei - 2020 - Studia Logica 108 (6):1161-1206.
    Hahn’s embedding theorem asserts that linearly ordered abelian groups embed in some lexicographic product of real groups. Hahn’s theorem is generalized to a class of residuated semigroups in this paper, namely, to odd involutive commutative residuated chains which possess only finitely many idempotent elements. To this end, the partial lexicographic product construction is introduced to construct new odd involutive commutative residuated lattices from a pair of odd involutive commutative residuated lattices, and a representation theorem for odd (...)
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  46.  11
    Decidability for ℤ2G‐lattices when G Extends the Noncyclic Group of Order 4.Annalisa Marcja & Carlo Toffalori - 2002 - Mathematical Logic Quarterly 48 (2):203-212.
    Let G be the direct sum of the noncyclic groupof order four and a cyclic groupwhoseorderisthe power pn of some prime p. We show that ℤ2G-lattices have a decidable theory when the cyclotomic polynomia equation image is irreducible modulo 2ℤ for every j ≤ n. More generally we discuss the decision problem for ℤ2G-lattices when G is a finite group whose Sylow 2-subgroups are isomorphic to the noncyclic group of order four.
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  47.  15
    Abelian‐by‐G Groups, for G Finite, from the Model Theoretic Point of View.Annalisa Marcja & Carlo Toffalori - 1994 - Mathematical Logic Quarterly 40 (1):125-131.
    Let G be a finite group. We prove that the theory af abelian-by-G groups is decidable if and only if the theory of modules over the group ring ℤ[G] is decidable. Then we study some model theoretic questions about abelian-by-G groups, in particular we show that their class is elementary when the order of G is squarefree.
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  48. The Nachtigall Convolute: A Previously Unknown Ottoman Protocol, Turkish Practices in the 1940s, and Possible Links between the Order of the Third Bird and the Work of Erich Auerbach.The Niblach Working Group - 2021 - In D. Graham Burnett, Catherine L. Hansen & Justin E. H. Smith (eds.), In search of the third bird: exemplary essays from the proceedings of ESTAR(SER), 2001-2021. London: Strange Attractor Press.
     
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  49. VI. Remains : The Finding Aid Folder: Seeking Order in the Archives of the Order.The Meta-Archival Working Group - 2021 - In D. Graham Burnett, Catherine L. Hansen & Justin E. H. Smith (eds.), In search of the third bird: exemplary essays from the proceedings of ESTAR(SER), 2001-2021. London: Strange Attractor Press.
     
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  50.  5
    Decidability for ℤ2 G-lattices when G Extends the Noncyclic Group of Order 4.Annalisa Marcja & Carlo Toffalori - 2002 - Mathematical Logic Quarterly 48 (2):203-212.
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